Key takeaway: The volume of a cone equals one third times pi times the radius squared times the height, exactly a third of the can that would snugly contain it.
Grade 88.G.C.9Geometry and Measurement
Definition of Volume of a Cone
The volume of a cone measures the space inside a solid that narrows from a circular base to a single point called the apex. Gravel piles, party hats, funnels and ice cream cones are all cones.
Fill a cone with water and empty it into a can of the same width and height: three pours are needed to fill the can. That experiment explains the one-third in the formula. Height means the plumb-line drop from the tip to the middle of the base, never the slanted edge.
Formula and notation
Let \(r\) be the radius of the base and \(h\) the perpendicular height:
\[V = \frac{1}{3}\pi r^{2} h\]
The part \(\pi r^{2}\) is the area of the circular base. The answer is in cubic units, for example cubic feet when \(r\) and \(h\) are in feet.
Key facts
- Three matching cones of water fill one can of equal width and height.
- The slant height is not used in the volume formula.
- Twice the height gives twice the volume of a cone; twice the radius gives four times .
- Give the radius, half the diameter, to the formula.
Volume of a Cone example
A conical gravel pile is 14 feet across at the ground and rises 6 feet at the peak. How many cubic feet of gravel are in the pile? Round to the nearest tenth.
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Radius: \(14 : 2 = 7\) feet.
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Substitute: \(V = \frac{1}{3}\pi (7)^{2}(6) = \frac{1}{3}\pi (49)(6)\).
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Simplify: \(\frac{49 \times 6}{3} = 98\), so \(V = 98\pi \approx 307.9\).
The pile holds about 307.9 cubic feet, exactly \(98\pi\) cubic feet.
Common mistake
Watch out
On the same pile, Noel leaves out the fraction and writes \(V = \pi (7)^{2}(6) = 294\pi \approx 923.6\) cubic feet.
Why it is wrong
\(\pi r^{2} h\) alone is the volume of a cylinder. A cone has only one third of that, so divide by 3: \(294\pi : 3 = 98\pi\). Compare with the cylinder: if the answer for the cone equals the cylinder’s, something is missing.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Geometry and Measurement